Stijnen et al. (2010)

The Methods and Data

Dichotomous and event count data (based on which one can calculate effect size or outcome measures, such as odds ratios, incidence rate ratios, proportions, and incidence rates) are often assumed to arise from binomial and Poisson distributed data. Stijnen et al. (2010) describe meta-analytic models that are directly based on these distibutions. The collection of models are essentially special cases of generalized linear (mixed-effects) models (i.e., mixed-effects logistic and Poisson regression models). For 2×2 table data, a mixed-effects conditional logistic model (based on the non-central hypergeometric distribution) can also be used. Mixed-effects logistic regression models for dichotomous data are often referred to as "binomial-normal models" in the meta-analytic literature. Analogously, mixed-effects Poisson regression models for event count data could be referred to as "Poisson-normal models". The results obtained from such models can be contrasted with those obtained from the standard (inverse-variance) approach (sometimes called the "normal-normal model"). The various models described in the article are illustrated with datasets from two meta-analyses comparing the risk of catheter-related bloodstream infection (CRBSI) when using anti-infective-treated versus standard catheters (Niel-Weise et al., 2007, 2008).

Binomial-Normal Model for the Meta-Analysis of Proportions

Niel-Weise et al. (2007) conducted a meta-analysis comparing the CRBSI risk when using anti-infective-treated versus standard catheters in the acute care setting. These data can be used to illustrate the binomial-normal model for the meta-analysis of proportions (i.e., infection risks) within the standard catheter and anti-infective catheter groups. The data can be extracted from Figure 2 (p. 2063) in Niel-Weise et al. (2007) and can be loaded with:

Variables n1i and n2i denote the number of patients receiving an anti-infective or standard catheter, respectively, and ai and ci the number of CRBSIs in the respective groups. We can now fit a "normal-normal model" to the data for the two groups separately. Here, we use the logit transformed proportions (i.e., log odds) for the meta-analysis. First, for the standard catheter groups, this can be done with:

As pointed out in the article, the proper interpretation of the back-transformed value (i.e., $0.036$) is that it reflects the median infection risk (although it would often be interpreted as the average risk).

Note that REML estimation is the default for the rma() function. Also, calculation of the log odds (and the corresponding sampling variance) is problematic when a group has zero events, as is the case in some of the studies. The standard continuity correction (adding 1/2 to the number of patients with and without an event) is automatically applied inside the function. These results match what is reported by Stijnen et al. (2010) in Table II (p. 3050).

Instead of the normal-normal model, we can use a "binomial-normal model", which in this case is just a logistic regression model with a random intercept. This model can be fitted to the data for the standard catheter groups with:

By default, a random/mixed-effects model using ML estimation is fitted by rma.glmm() as was done in the article. Again, these results match what is reported in the article.

Hypergeometric-Normal Model for the Meta-Analysis of Odds Ratios

Next, the authors describe a "hypergeometric-normal model" for the meta-analysis of odds ratios. This is essentially a mixed-effects conditional logistic model. The model results from conditioning on the total number of cases within each study, leading to the non-central hypergeometric distribution for the 2×2 tables. Since fitting this model can be difficult and computationally expensive, one can approximate the exact likelihood by a binomial distribution, which works well when the number of cases in each study is small relative to the group sizes (as is the case here). Again, the results obtained from this model can be compared with the normal-normal model approach.

In fact, again, we start with the normal-normal model, using the log odds ratio as the outcome measure. This model can be fitted with:

Poisson-Normal Model for the Meta-Analysis of Incidence Rates

Niel-Weise et al. (2008) conducted a meta-analysis comparing the CRBSI risk when using anti-infective-treated versus standard catheters for total parenteral nutrition or chemotherapy. These data can be used to illustrate Poisson-normal models for the meta-analysis of incidence rates and incidence rate ratios. The data can be extracted from Table III (p. 119) in Niel-Weise et al. (2008) and can be loaded with:

Here, the number of infections (x1i and x2i) and the total number of catheter days (t1i and t2i) are given for patients in the anti-infective-treated and standard catheter groups, respectively. Note that these data differ very slighty (for a few studies) from the data reported by Stijnen et al. (2010) in the appendix of their article (p. 3062)1) but these differences are inconsequential for the results.

Following the authors, we first divide the total number of catheter days by 1000, so that the estimated (average) incidence rates reflect the expected number of events per 1000 days:

dat$t1i <- dat$t1i/1000
dat$t2i <- dat$t2i/1000

Next, the normal-normal model can be fitted using the log incidence rate as the outcome measure. For patients in the standard catheter groups, this can be done with:

Binomial-Normal Model for the Meta-Analysis of Incidence Rate Ratios

Next, we consider the mixed-effects conditional Poisson regression model described by the authors. Through the conditioning on the total number of events within each study, we actually obtain a binomial-normal model for the data within each study. But first, we start again with the normal-normal model, using the log incidence rate ratio as the outcome measure:

And in fact, these data differ from what is reported in Table VI (p. 3054) in the same article. However, Stijnen et al. (2010) clearly used the data from the appendix for the analysis, so Table VI can be ignored.